BIZOMAT MAKEOS · Mathematics
Complex Number Calculator
What is the sum, difference, product or quotient of these complex numbers, in rectangular and polar form?
Add, subtract, multiply and divide complex numbers and convert them between rectangular (a + bi) and polar (magnitude and angle) form, for AC circuits, vibration and phasor work.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Operation | choice | Operation to perform. |
| z₁ real part | Real part of the first number. | |
| z₁ imaginary part | Imaginary part of the first number. | |
| z₂ real part | Real part of the second number. | |
| z₂ imaginary part | Imaginary part of the second number. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Operation
- What it is
- The arithmetic to carry out on the two complex numbers z₁ and z₂: add, subtract, multiply or divide.
- Units and format
- Choose one of 4 options from the list: Add (z₁ + z₂); Subtract (z₁ − z₂); Multiply (z₁ × z₂); Divide (z₁ ÷ z₂).
- Example
- The form opens with “Multiply (z₁ × z₂)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Subtract and divide take z₁ first: the result is z₁ − z₂ or z₁ ÷ z₂.
What this means: z₁ real part
- What it is
- The real part of the first complex number z₁ = a + bi, the part with no i attached.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For a number given in polar form, real part = magnitude × cos(angle).
What this means: z₁ imaginary part
- What it is
- The imaginary part of the first complex number z₁ = a + bi, the number multiplying i.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type the coefficient only, with its sign: for 3 − 4i type −4.
What this means: z₂ real part
- What it is
- The real part of the second complex number z₂ = c + di, the part with no i attached.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Division needs z₂ to be non-zero, so c and d cannot both be 0.
What this means: z₂ imaginary part
- What it is
- The imaginary part of the second complex number z₂ = c + di, the number multiplying i.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type the coefficient only, with its sign: for 1 − 2i type −2.
Method and formulas
- (a + bi)(c + di) = (ac − bd) + (ad + bc)i; (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²).
- Polar: magnitude √(re² + im²), angle atan2(im, re).
- Example: (3 + 4i) × (1 -2i) = 11 + -2i.
Limitations
- Angles are reported in degrees from −180° to 180°.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Combine impedances and phasors
- Convert between rectangular and polar form
- Check vibration and control calculations
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- Result in rectangular form
- Result in polar form
- Polar forms of the inputs
Interpreting the result
- Multiply magnitudes and add angles to check products in polar form.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 18–19. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Dielectric loss and tan delta
In Insulation Testing by the Numbers
What it covers: AC tests of insulation: complex permittivity, the loss angle and tan delta, C-1's charging current and loss, relaxation peaks, capacitance and moisture, tip-up, and an alarm raised by a warm winding.
Method explained here · BIZOMAT original book
Unbalance and balancing in place
In Vibration Monitoring by the Numbers
What it covers: How unbalance makes a 1× vibration, the force on the bearings, balance quality grades, single-plane balancing with a trial weight and vectors, splitting a correction, and a trial weight left on.
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